$\dfrac{ 7f - 8g }{ 10 } = \dfrac{ 10f - 10h }{ 4 }$ Solve for $f$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ 7f - 8g }{ {10} } = \dfrac{ 10f - 10h }{ 4 }$ ${10} \cdot \dfrac{ 7f - 8g }{ {10} } = {10} \cdot \dfrac{ 10f - 10h }{ 4 }$ $7f - 8g = {10} \cdot \dfrac { 10f - 10h }{ 4 }$ Multiply both sides by the right denominator. $7f - 8g = 10 \cdot \dfrac{ 10f - 10h }{ {4} }$ ${4} \cdot \left( 7f - 8g \right) = {4} \cdot 10 \cdot \dfrac{ 10f - 10h }{ {4} }$ ${4} \cdot \left( 7f - 8g \right) = 10 \cdot \left( 10f - 10h \right)$ Distribute both sides ${4} \cdot \left( 7f - 8g \right) = {10} \cdot \left( 10f - 10h \right)$ ${28}f - {32}g = {100}f - {100}h$ Combine $f$ terms on the left. ${28f} - 32g = {100f} - 100h$ $-{72f} - 32g = -100h$ Move the $g$ term to the right. $-72f - {32g} = -100h$ $-72f = -100h + {32g}$ Isolate $f$ by dividing both sides by its coefficient. $-{72}f = -100h + 32g$ $f = \dfrac{ -100h + 32g }{ -{72} }$ All of these terms are divisible by $4$ Divide by the common factor and swap signs so the denominator isn't negative. $f = \dfrac{ {25}h - {8}g }{ {18} }$